The last temptation of William T. Tutte

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Abstract

In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times of the Four Color Conjecture. He asked whether the 4-coloring complex of a planar triangulation could have two components in which all colorings had the same parity. In this note we answer Tutte’s question contrary to his speculations by showing that there are triangulations of the plane whose coloring complexes have arbitrarily many even and odd components.

Introduction

In [3], Tutte asked if there existed a triangulation of the plane whose 4-coloring complex had more than one component of the same parity. He based this question on numerous examples he examined over the course of many years. One obvious possibility is the icosahedron, whose 4-colorings form ten Kempe equivalence classes. However, Tutte found out that the icosahedron has a connected coloring complex. Nonetheless, he was resistant to making this question into a conjecture that no such examples exist, since, in his view, “the data are too few to justify a Conjecture”. However, he asked: “If anyone knows of any case of two components of the same parity, I would be glad to hear of it”.

In this short note, we provide examples of 4-connected triangulations of the plane whose 4-coloring complexes have arbitrarily many components with odd colorings and arbitrarily many components with even colorings. Although this seems to resolve the Tutte problem, we end up with a closely related conjecture, which is based on an extensive computation, and which claims that for every planar triangulation whose 4-coloring complex is disconnected has a component of even parity and one of odd parity (see Conjecture 8).

Section snippets

Coloring complexes and the parity of 4-colorings

Given a 4-colorable graph G, an independent set of vertices AV(G) is said to be a color class if it is one of the color classes for some 4-coloring of G. The 4-coloring complex B(G) is the graph which has all the color classes of G as its vertices and two vertices C,DV(B(G)) joined by an edge if the color classes C and D appear together in a 4-coloring of G. These graphs were introduced by Tutte [2], who discussed their basic structure. See Fig. 2 for an example of a coloring complex.

A

4-colorings of triangulated surfaces

Fisk [1] introduced an alternative view of the parity of 4-colorings. We show how to reconcile this approach with Tutte’s definition.

Given a triangulation T of an orientable surface with one of its orientations fixed, we can view a 4-coloring f of T as a simplicial mapping onto the boundary of the tetrahedron, which will be denoted as K4. For each triangle Ti,j,l of K4, we consider the facial triangles in T that are mapped onto Ti,j,l in such a way that their orientation is preserved and those

Coloring complexes with many components

We have found two triangulations of the plane on 12 vertices which answer Tutte’s question in the affirmative. Their 4-coloring complexes each have three components, from which it follows that two of these components must have the same parity. These triangulations are drawn in Fig. 1.

We will now determine the 4-coloring complex of Example 1, and show that it has 3 connected components (the argument for Example 2 is similar).

Theorem 4

The 4-coloring complex of the first example G given in Fig. 1(a) is the

4-connected examples

Identification over a triangle produces triangulations that are not 4-connected. However, it may be that in the back of Tutte’s mind was an implicit condition that the triangulations should be4-connected. The following generalized examples show that one can also find 4-connected examples whose coloring complexes have many components (of the same parity). The two graphs Q1 and Q1 in Fig. 1 are just smallest examples in two infinite families of 4-connected triangulations. We define Q0,Q1,Q2, as

Next steps

At present, we have not found 5-connected triangulations of the plane whose 4-coloring complexes would have arbitrarily many components. However, we have found some 5-connected examples whose 4-coloring complexes have more than one component of the same parity. The three smallest triangulations of this kind are illustrated in Fig. 4. These are triangulations of the plane on 20 vertices, and their minimality follows from an extensive computation that we performed on 5-connected triangulations

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1

Supported in part by an NSERC Discovery Grant R611450 (Canada), by the Canada Research Chair program, and by the Research Grant J1-8130 of ARRS (Slovenia) .

2

On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.

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